Why are the bid prices of T-bills higher than the ask prices? Aren't bids supposed to be lower than ask prices?

Yes, you are correct that the ask price of a security should typically be higher than the bid price. This is because people will not sell a security (asking price) for lower than the price they are willing to pay for it (bidding price). So, because there is more than one method of quoting the bid and ask prices of T-bills, the quoted ask price may simply be perceived as being lower than the bid.

For example, one common quote that you may see for a 365-day T-bill is July 12th, bid 5.35%, ask 5.25%. At first glance, the bid seems higher than the ask, but upon further inspection, you can see that the ask is actually higher. The reason is that a T-bill is a discount bond and these percentages are the quoted yields, not the actual prices. So, if we convert the bid and ask discount yields into the dollar amounts of the prices, we get a bid of $94.65 and an ask of $94.75. Therefore, the bid is actually lower than the ask. Sometimes the quotes on T-bills show the actual prices, in which case you don't have to convert or calculate anything. The same T-bill above, therefore, may be quoted with a bid of 94.65 and an ask of 94.75.

So, as the dollar amount of the bid should be lower than the ask, the bid's quoted yield percentage should be higher than the ask's quoted yield percentage - the two different kinds of quotes are just different ways of saying the same thing.